I'm an electrician. There is very little power available through the lighter. If you were serious, run new feeds from the battery. Use at least a #4awg and run to a dedicated inverter, say a 500w or more. I can always crunch the numbers for you and size your conductors
Discussion
The Bitaxe gamma uses 21.6 watts. I am planning to use an AC plug in an old car, that came with it. It is rated for 150 watts maximum! It should be good, I think!